Momentum Noncommutativity Research Articles

On the entanglement of co-ordinate and momentum degrees of freedom in noncommutative space

In this paper, we investigate the quantum entanglement induced by phase-space noncommutativity. Both the position–position and momentum–momentum noncommutativity are incorporated to study the entanglement properties of coordinate and momentum degrees of freedom under the shade of oscillators in noncommutative space. Exact solutions for the systems are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommuting degrees of freedom. It is shown that the bipartite Gaussian state for an isotropic oscillator is always separable. To extend our study for the time-dependent system, we allow arbitrary time dependency on parameters. The time-dependent isotropic oscillator is solved with the Lewis–Riesenfeld invariant method. It turns out that even for arbitrary time-dependent scenarios, the separability property does not alter. We extend our study to the anisotropic oscillator, which provides an entangled state even for time-independent parameters. The Wigner quasi-probability distribution is constructed for a bipartite Gaussian state. The noise matrix (covariance matrix) is explicitly studied with the help of Wigner distribution. Simon’s separability criterion (generalized Peres–Horodecki criterion) has been employed to find the unique function of the (mass and frequency) parameters, for which the bipartite states are separable. In particular, we show that the mere inclusion of non-commutativity of phase-space is not sufficient to generate the entanglement, rather anisotropy is important at the same footing. We explore the experimental viability of our result through the computation of extractable work for the current situation.

Entanglement in phase-space distribution for an anisotropic harmonic oscillator in noncommutative space

The bipartite Gaussian state, corresponding to an anisotropic harmonic oscillator in a noncommutative space (NCS), is investigated with the help of Simon’s separability condition (generalized Peres-Horodecki criterion). It turns out that, to exhibit the entanglement between the noncommutative co-ordinates, the parameters (mass and frequency) have to satisfy a unique constraint equation. We have considered the most general form of an anisotropic oscillator in NCS, with both spatial and momentum noncommutativity. The system is transformed to the usual commutative space (with usual Heisenberg algebra) by a well-known Bopp shift. The system is transformed into an equivalent simple system by a unitary transformation, keeping the intrinsic symplectic structure (\(Sp(4,{\mathbb {R}})\)) intact. Wigner quasiprobability distribution is constructed for the bipartite Gaussian state with the help of a Fourier transformation of the characteristic function. It is shown that the identification of the entangled degrees of freedom is possible by studying the Wigner quasiprobability distribution in phase space. We have shown that the co-ordinates are entangled only with the conjugate momentum corresponding to other co-ordinates.

Emergent time crystals from phase-space noncommutative quantum mechanics

It has been argued that the existence of time crystals requires a spontaneous breakdown of the continuous time translation symmetry so to account for the unexpected non-stationary behavior of quantum observables in the ground state. Our point is that such effects do emerge from position (qˆi) and/or momentum (pˆi) noncommutativity, i.e., from [qˆi,qˆj]≠0 and/or [pˆi,pˆj]≠0 (for i≠j). In such a context, a predictive analysis is carried out for the 2-dim noncommutative quantum harmonic oscillator through a procedure supported by the Weyl-Wigner-Groenewold-Moyal framework. This allows for the understanding of how the phase-space noncommutativity drives the amplitude of periodic oscillations identified as time crystals. A natural extension of our analysis also shows how the spontaneous formation of time quasi-crystals can arise.

Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators

Without access to the full quantum state, modeling quantum transport in mesoscopic systems requires dealing with a limited number of degrees of freedom. In this work, we analyze the possibility of modeling the perturbation induced by non-simulated degrees of freedom on the simulated ones as a transition between single-particle pure states. First, we show that Bohmian conditional wave functions (BCWFs) allow for a rigorous discussion of the dynamics of electrons inside open quantum systems in terms of single-particle time-dependent pure states, either under Markovian or non-Markovian conditions. Second, we discuss the practical application of the method for modeling light–matter interaction phenomena in a resonant tunneling device, where a single photon interacts with a single electron. Third, we emphasize the importance of interpreting such a scattering mechanism as a transition between initial and final single-particle BCWF with well-defined central energies (rather than with well-defined central momenta).

Squeezed coherent states for gravitational well in noncommutative space

Gravitational well is a widely used system for the verification of the quantum weak equivalence principle (WEP). We have studied the quantum gravitational well under the shed of noncommutative (NC) space so that the results can be utilized for testing the validity of WEP in NC-space. To keep our study widely usable, we have considered both position–position and momentum–momentum noncommutativity. Since coherent state (CS) structure provides a natural bridge between the classical and quantum domain descriptions, the quantum domain validity of purely classical phenomena like free fall under gravity might be verified with the help of CS. We have constructed CS with the aid of a Lewis–Riesenfeld phase-space invariant operator. We deduced the uncertainty relations from the expectation values of the observables and showed that the solutions of the time-dependent Schrödinger equation are squeezed coherent states.

Angular momentum quantum backflow in the noncommutative plane

We study the quantum backflow problem in the noncommutative plane. In particular, we have considered a charged particle with and without an oscillator interaction with noncommuting momentum operators and examined angular momentum backflow in each case and how they differ from each other. We also propose a probability associated with the occurence of angular momentum backflow and investigate whether or not the probability depends on a physical parameter, namely the magnetic field.

Quantum Analysis of BTZ Black Hole Formation Due to the Collapse of a Dust Shell

We perform Hamiltonian reduction of a model in which 2 + 1 dimensional gravity with negative cosmological constant is coupled to a cylindrically symmetric dust shell. The resulting action contains only a finite number of degrees of freedom. The phase space consists of two copies of ADS2—both coordinate and momentum space are curved. Different regions in the Penrose diagram can be identified with different patches of ADS2 momentum space. Quantization in the momentum representation becomes particularly simple in the vicinity of the horizon, where one can neglect momentum non-commutativity. In this region, we calculate the spectrum of the shell radius. This spectrum turns out to be continuous outside the horizon and becomes discrete inside the horizon with eigenvalue spacing proportional to the square root of the black hole mass. We also calculate numerically quantum transition amplitudes between different regions of the Penrose diagram in the vicinity of the horizon. This calculation shows a possibility of quantum tunneling of the shell into classically forbidden regions of the Penrose diagram, although with an exponentially damped rate away from the horizon.

Emergence of a geometric phase shift in planar noncommutative quantum mechanics

Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of 2D simple harmonic oscilator in Moyal plane, where momentum non-commutativity are also considered along with spatial noncommutativity. After finding a suitable Bopp shift, that bridges the noncommutative phase space operators with their effective commutative counterparts, having their dependence on the non-commutative parameters, we study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to the Hannay angle of a classically analogous system, by studying the evolution of the coherent state of this system.

Minimal momentum estimation in noncommutative phase space of canonical type with preserved rotational and time reversal symmetries

Noncommutative algebra which is rotationally invariant, time reversal invariant and equivalent to noncommutative algebra of canonical type is considered. Perihelion shift of orbit of a particle in Coulomb potential in the rotationally-invariant noncommutative phase space is found up to the second order in the parameters of noncommutativity. Applying the result to the case of Mercury planet and using observable results for precession of its orbit we find upper bounds on the parameters of noncommutativity in the rotationally-invariant noncommutative phase space. The obtained upper bound for the parameter of momentum noncommutativity is at least ten orders less than the upper bounds estimated on the basis of studies of the hydrogen atom in noncommutative phase space. As a result we obtain stringent restriction on for the minimal momentum in noncommutative phase space with preserved rotational and time reversal symmetries.

Noncommutative Momentum and Torsional Regularization

We show that in the presence of the torsion tensor $S^k_{\phantom{k}ij}$, the quantum commutation relation for the four-momentum, traced over spinor indices, is given by $[p_i,p_j]=2i\hbar S^k_{\phantom{k}ij}p_k$. In the Einstein--Cartan theory of gravity, in which torsion is coupled to spin of fermions, this relation in a coordinate frame reduces to a commutation relation of noncommutative momentum space, $[p_i,p_j]=i\epsilon_{ijk}Up^3 p_k$, where $U$ is a constant on the order of the squared inverse of the Planck mass. We propose that this relation replaces the integration in the momentum space in Feynman diagrams with the summation over the discrete momentum eigenvalues. We derive a prescription for this summation that agrees with convergent integrals: \[ \int\frac{d^4p}{(p^2+\Delta)^s}\rightarrow 4\pi U^{s-2}\sum_{l=1}^\infty \int_0^{\pi/2} d\phi \frac{\sin^4\phi\,n^{s-3}}{[\sin\phi+U\Delta n]^s}, \] where $n=\sqrt{l(l+1)}$ and $\Delta$ does not depend on $p$. We show that this prescription regularizes ultraviolet-divergent integrals in loop diagrams. We extend this prescription to tensor integrals. We derive a finite, gauge-invariant vacuum polarization tensor and a finite running coupling. Including loops from all charged fermions, we find a finite value for the bare electric charge of an electron: $\approx -1.22\,e$. This torsional regularization may therefore provide a realistic, physical mechanism for eliminating infinities in quantum field theory and making renormalization finite.

Photon Gas Thermodynamics in Noncommutative Momentum Spaces

In this paper, we study thermodynamical properties of the extreme relativistic gases in canonical ensembles in non-commutative momentum spaces. In this regards, we exactly calculate the modified partition function, Helmholtz free energy, internal energy, entropy, heat capacity and the thermal pressure in the momentum spaces. Moreover, we conclude at high temperature limits due to the decreasing of the number of micro-states, these quantities reach to maximal bounds which do not exist in standard cases and it concludes that at the presence of gravity for both micro-canonical and canonical ensembles, the internal energy and the entropy tend to these upper bounds.

Upper bound on the momentum scale in noncommutative phase space of canonical type

We find a stringent upper bound on the momentum scale in noncommutative phase space of canonical type on the basis of studies of the perihelion shift of the Mercury planet by taking into account features of the description of the motion of a macroscopic body in the space with noncommutativity of coordinates and noncommutativity of momenta. Using results for precession of the perihelion of the Mercury planet from ranging to the MESSENGER spacecraft we obtain an upper bound for the parameter of momentum noncommutativity 10-83 kg2m2/s2 which is many orders less than known in the literature.

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.

Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle

We study the influence of noncommutativity of coordinates and noncommutativity of momenta on the motion of a particle (macroscopic body) in uniform and nonuniform gravitational fields in noncommutative phase space of canonical type with preserved rotational symmetry. It is shown that because of noncommutativity the motion of a particle in a gravitational field is determined by its mass. The trajectory of motion of a particle in a uniform gravitational field corresponds to the trajectory of a harmonic oscillator with frequency determined by the value of the parameter of momentum noncommutativity and mass of the particle. The equations of motion of a macroscopic body in a gravitational field depend on its mass and composition. From this follows a violation of the weak equivalence principle caused by noncommutativity. We conclude that the weak equivalence principle is recovered in rotationally invariant noncommutative phase space if we consider the tensors of noncommutativity to be dependent on mass. So, finally we construct noncommutative algebra which is rotationally invariant, equivalent to noncommutative algebra of canonical type, and does not lead to violation of the weak equivalence principle.

Deformations of Fluid Dynamics and Friedmann Equation on Noncommutative Phase Space

Noncommutative phase space is one of the widely studied extensions of ordinary phase space, and has profound implications in cosmological physics. In this paper we study the dynamics of perfect fluid on noncommutative phase space, as well as deformations of the Friedmann equation. The Lagrangian formalism is used to take into account of the phase space noncommutativities. Then a map from canonical Lagrangian variables to Eulerian variables is employed to derive the equations of motion of the mass and current densities. We find that both these two equations receive noncommutative corrections that are linear in the noncommutative parameters. However, we also find that in the approximation of vanishing comoving velocity the leading order noncommutative correction due to momentum noncommutativity on the Friedmann equation is zero.

Influence of Noncommutativity on the Motion of Sun-Earth-Moon System and the Weak Equivalence Principle

Features of motion of macroscopic body in gravitational field in a space with noncommutativity of coordinates and noncommutativity of momenta are considered in general case when coordinates and momenta of different particles satisfy noncommutative algebra with different parameters of noncommutativity. Influence of noncommutativity on the motion of three-body Sun-Earth-Moon system is examined. We show that because of noncommutativity the free fall accelerations of the Moon and the Earth toward the Sun in the case when the Moon and the Earth are at the same distance to the source of gravity are not the same even if gravitational and inertial masses of the bodies are equal. Therefore, the Eotvos-parameter is not equal to zero and the weak equivalence principle is violated in noncommutative phase space. We estimate the corrections to the Eotvos-parameter caused by noncommutativity on the basis of Lunar laser ranging experiment results. We obtain that with high precision the ratio of parameter of momentum noncommutativity to mass is the same for different particles.

Features of free particles system motion in noncommutative phase space and conservation of the total momentum

Influence of noncommutativity on the motion of composite system is studied in noncommutative phase space of canonical type. A system composed by N free particles is examined. We show that because of the momentum noncommutativity free particles of different masses with the same velocities at the initial moment of time do not move together. The trajectory and the velocity of a free particle in noncommutative phase space depend on its mass. So, a system of the free particles flies away. Also, it is shown that the total momentum defined in the traditional way is not integral of motion in a space with noncommutativity of coordinates and noncommutativity of momenta. We find that in the case when parameters of noncommutativity corresponding to a particle are determined by its mass, the trajectory and the velocity of the free particles are independent of the mass. Also, the total momenta as integrals of motion can be introduced in noncommutative phase space.

On a causal quantum stochastic double product integral related to Lévy area

We study the family of causal double product integrals \prod_{a < x < y < b}\Big(1 + i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y + dQ_x dQ_y)\Big), where P and Q are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in [15]. The main problem solved in this paper is the explicit evaluation of the continuum limit W of the latter, and showing that W is a unitary operator. The kernel of W - I is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.

Non-commutativity effects in the Dirac equation in crossed electric and magnetic fields

In this paper we present exact solutions of the Dirac equation on the non-commutative plane in the presence of crossed electric and magnetic fields. In the standard commutative plane such a system is known to exhibit contraction of Landau levels when the electric field approaches a critical value. In the present case we find exact solutions in terms of the non-commutative parameters η (momentum non-commutativity) and θ (coordinate non-commutativity) and provide an explicit expression for the Landau levels. We show that non-commutativity preserves the collapse of the spectrum. We provide a dual description of the system: i) one in which at a given electric field the magnetic field is varied and the other ii) in which at a given magnetic field the electric field is varied. In the former case we find that momentum non-commutativity (η) splits the critical magnetic field into two critical fields while coordinates non-commutativity (θ) gives rise to two additional critical points not at all present in the commutative scenario.

Probing noncommutativities of phase space by using persistent charged current and its asymmetry

We study the physical properties of the persistent charged current in a metal ring on a noncommutative phase space, and temperature dependence of the noncommutative corrections are also analyzed. We find that the coordinate noncommutativity only affects the total magnitude of the current, and it is difficult to observe it. In contrast, the momentum noncommutativity can violate symmetry property of the current, and this violation of symmetry holds at a finite temperature. Based on this violation effect, we introduce an asymmetry observable to measure the momentum noncommutativity.